Heat conduction in a thin circular pipe due to a circumferentially moving heat source

The solution of the title problem is often approximated by the temperature distribution in a flat plate due to an array of periodically placed heat sources, where the curvature of the pipe is neglected. Here, using an asymptotic expansion in terms of the thickness to radius ratio, linear corrections to the flat plate solution are obtained explicitly. Results indicate that these corrections may be significant depending, of course, on the above small parameter and the speed of the moving source.